In this post I’ll be demonstrating how you can prove that **Thales’ Theorem** is true. To follow the steps in this post (11 in total), what you will require is a *ruler, pair of compasses and a pencil*.

##### Step 1: Draw a random line on a sheet of paper.

##### Step 2: Place your compass needle on this line, and form a circle.

##### Step 3: Add 4 points to your drawing, as shown below…

##### Step 4: Name the points A, B, C and D as shown…

##### Step 5: Connect the points A, B and D together to form an **isosceles triangle**…

##### Step 6: Name the lines AB and BD the radius (r)…

##### Step 7: Since the lines AB and BD are equal to one another, it follows that the angles ∠BAD and ∠BDA are equivalent. This is because **the angles below the apex of an isosceles triangle are equal**. You must name these angles alpha (α).

##### Step 8: Now connect the points BC and CD together to form another isosceles triangle…

##### Step 9: The line BC is equal to r… Now label the line BC…

##### Step 10: Because the line BC and BD are both equal to r, the triangle BCD is an isosceles triangle. This means that the angles ∠BCD and ∠BDC must both be equivalent. Call these angles beta (β).

##### Step 11: Prove that the angle at point D is equal to 90 degrees.

*Thales’ Theorem is as follows:*

*Because AC is the diameter of the circle you drew, the angle at the point D (α+β) must be equal to 90 degrees. In more specific and general terms, if you have the points A, C and D lying on a circle – and the line AC is in fact the diameter of this circle – then the angle at point D (α+β) must be a right angle.*

**Proof (which must be derived using the diagram you’ve created):**

All angles within a triangle (in 2 space) must add up to 180 degrees.

Mathematically, this means that:

And as a result, Thales’ theorem **must be true**. The angle **α+β** is the angle at point **D**.